Square 2+ Digit Numbers Mentally

Step 2 Strategies for Computing Squares

Alright, hopefully at this point we've either practiced or ignored the previous step and clicked ahead anyway. Suppose we're given a number, such as 45. How can we square the number mentally?

Turns out there are a couple different ways:
1) Memorize every 2-digit number, and simply recite the square from memory (rote memorization).
2) Multiply the number by itself mentally (45*45)
3) Split 45 into components and apply the identity (a+b)² = a²+2ab+b²
4) Apply a high speed Vedic Mathematics technique for squaring numbers

As can be seen from above, the first method isn't general or practical. Even if you knew the squares of every 2 digit number, that's only 99 numbers (I'm including 1-9 here, I'm aware they are not 2-digits), and in the broader realm of math, won't get you very far.

Method 2 seems like a reasonable approach, though. However, the numbers you have to hold in memory are not very nice. At one time, you must remember that 5*45 is 225 and 40*45 is 1800 and add them in your head. For many numbers, this proves to be quite difficult.

Method 3 is a decent method, decidedly better than the first 2. We could express 45² in the following way:
45²= (40+5)²=40²+2*40*5+5² = 1600 + 400 + 25 = 2025

But, better than any of those is the 4th method, which I will show you next.